Non-local effects in the fermion dynamical mean-field framework; application to the two-dimensional Falicov-Kimball model

We propose a new, controlled approximation scheme that explicitly includes the effects of non-local correlations on the D = infinity solution. In cont rast to the usual D = infinity case, the self-energy is self-consistently c oupled to two-particle correlation functions. The formalism is general, and is applied to the two-dimensional Falicov-Kimball model. Our approach poss esses all the strengths of the large-D solution, and allows one to undertak e a systematic study of the effects of inclusion of k-dependent effects on the D = infinity picture. Results for the density of states rho(omega), and the single-particle spectral density for the 2D Falicov-Kimball model alwa ys yield positive definite rho(omega), and the spectral function shows stri king new features inaccessible for D = infinity. Our results are in good ag reement with the exact results known from the 2D Falicov-Kimball model.